A statistical analysis of single and multiple response surface modeling

Abstract

This work examines the use of single response surface (SRS) and multiple response surface (MRS) techniques for modeling spatial nonuniformity in semiconductor applications. Previous works have suggested that the MRS estimation techniques better measure the nonuniformity due to the underlying spatial function of the process, whereas SRS estimation methods measure the total process nonuniformity (systematic spatial nonuniformity plus random site nonuniformity). This work further highlights this fact in an analytical setting. It is demonstrated that the MRS estimation technique is biased and that this bias can lead to the choice of a nonoptimal process. Experimental data from a chemical-mechanical polishing (CMP) process confirms these observations and demonstrates that careful use of the MRS estimator is required in achieving meaningful results for estimating spatial nonuniformity. Modified versions of each method, which measure spatial nonuniformity alone, as well as versions which measure total nonuniformity, are proposed for the case when one is comparing discrete process settings. Analytical expressions for the expected value and variance of both the SRS and MRS estimators are determined. These are used to compare the efficiency (estimator variance) of these modified estimators. When comparing spatial nonuniformity, it is found that the unbiased MRS estimator is more efficient than the SRS estimator modified to measure spatial nonuniformity. However, it is shown that the MRS estimator, when modified to measure total nonuniformity, is not necessarily more efficient than the SRS method. Finally, the continuous response surface modeling case is considered. It is demonstrated how confidence intervals on the underlying continuous site models lead to a nonuniform bias in the response surface generated by the MRS method. This suggests that care must be taken when using the MRS technique in creating continuous response surfaces of spatial nonuniformity as a function of the process settings.